### Abstract

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: KE be a continuous pseudocontractive mapping and f:KE a contraction, both satisfying weakly inward condition. Then for t(0, 1), there exists a sequence {yt}K satisfying the following condition: yt=(1-t)f(yt)+tT(yt). Suppose further that {yt} is bounded or F(T) and E is a reflexive Banach space having weakly continuous duality mapping J for some gauge . Then it is proved that {yt} converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.

Original language | English |
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Pages (from-to) | 1405-1419 |

Number of pages | 15 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 28 |

Issue number | 11-12 |

DOIs | |

Publication status | Published - Jan 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

*Numerical Functional Analysis and Optimization*,

*28*(11-12), 1405-1419. https://doi.org/10.1080/01630560701749730

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*Numerical Functional Analysis and Optimization*, vol. 28, no. 11-12, pp. 1405-1419. https://doi.org/10.1080/01630560701749730

**Approximation methods for a common fixed point of a finite family of nonexpansive mappings.** / Zegeye, Habtu; Shahzad, Naseer.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximation methods for a common fixed point of a finite family of nonexpansive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2007/1/1

Y1 - 2007/1/1

N2 - Let K be a nonempty closed and convex subset of a real Banach space E. Let T: KE be a continuous pseudocontractive mapping and f:KE a contraction, both satisfying weakly inward condition. Then for t(0, 1), there exists a sequence {yt}K satisfying the following condition: yt=(1-t)f(yt)+tT(yt). Suppose further that {yt} is bounded or F(T) and E is a reflexive Banach space having weakly continuous duality mapping J for some gauge . Then it is proved that {yt} converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.

AB - Let K be a nonempty closed and convex subset of a real Banach space E. Let T: KE be a continuous pseudocontractive mapping and f:KE a contraction, both satisfying weakly inward condition. Then for t(0, 1), there exists a sequence {yt}K satisfying the following condition: yt=(1-t)f(yt)+tT(yt). Suppose further that {yt} is bounded or F(T) and E is a reflexive Banach space having weakly continuous duality mapping J for some gauge . Then it is proved that {yt} converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.

UR - http://www.scopus.com/inward/record.url?scp=37549055443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37549055443&partnerID=8YFLogxK

U2 - 10.1080/01630560701749730

DO - 10.1080/01630560701749730

M3 - Article

AN - SCOPUS:37549055443

VL - 28

SP - 1405

EP - 1419

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 11-12

ER -